Problem 133 - College Algebra Essentials

Question

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. The toll to a bridge is \(\$ 3.00 .\) A three-month pass costs \(\$ 7.50\) and reduces the toll to \(\$ 0.50 .\) A six-month pass costs \(\$ 30\) and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three month pass to be the best deal?

Step-by-Step Solution

Answer

The three-month pass becomes the best deal for crossing the bridge when one has to make more than three crossings within a three-month period.

Step 1: Formulate the cost function for each scenario

For a regular toll without pass, the cost per crossing is $3. So, the total cost of crossing the bridge n times would be 3n. The cost for a three-month pass is $7.5 and the cost per crossing is reduced to $0.5. Total cost would therefore be 7.5 + 0.5n. A six-month pass costs $30 with no additional fee per crossing. Hence the cost is constant at 30.

Step 2: Set up the inequality

Since the goal is to find out the number of crossings for which the three-month pass is the best deal, an inequality needs to be set up comparing the cost of three-month pass and the cost of daily pass. Therefore, the inequality is: \(3n > 7.5 + 0.5n\)

Step 3: Solving the inequality

First simplify the above inequality by subtracting \(0.5n\) from both sides: \(2.5n > 7.5\). Dividing both sides by 2.5 gives: \(n >3\)

Step 4: Interpret the result

The inequality \(n > 3\) implies that the three-month pass becomes a better deal when one needs to cross the bridge more than three times within a three-month period.

Key Concepts

Word Problems

Word problems are often used in math to represent real-life situations through mathematical expressions. Solving word problems involves understanding the problem statement, identifying the necessary information, and translating the word problem into a mathematical expression or equation.
To successfully solve the given bridge toll problem, one must interpret the scenarios:

• The regular toll costs $3 per crossing.
• A three-month pass costs $7.50, reducing the toll to $0.50 per crossing.
• A six-month pass costs $30 and eliminates per-crossing fees.

These costs need to be compared to determine which pass offers the best value based on crossing frequency. This requires translating the problem into a mathematical model using linear inequalities.

Cost Function

The cost function helps to calculate expenses associated with different options based on variable conditions — in this case, the number of bridge crossings. A cost function represents this relationship mathematically.
For the regular toll without a pass, the cost function is simple:

• Total cost = \(3 * n, where \(n\) is the number of crossings.

For the three-month pass, it combines a fixed cost with a variable cost:

• Total cost = \)7.50 + \(0.50 * n

Lastly, the six-month pass incurs a flat fee no matter how many crossings you make:

• Total cost = \)30

Understanding these cost functions is crucial for comparing options effectively.

Solving Inequalities

Solving inequalities involves finding the values of a variable that make one expression greater than or less than another. In the context of this problem, the inequality helps identify when one toll option is cheaper than the other.
The inequality to determine when the three-month pass is a better deal than paying the regular toll is:

• \(3n > 7.5 + 0.5n\)

Simplifying this involves:

• Subtract \(0.5n\) from both sides: \(2.5n > 7.5\)
• Dividing by 2.5: \(n > 3\)

Hence, more than three crossings per three-month period make the pass worthwhile. This solution shows how inequalities can help in decision-making processes.

Mathematical Modeling

Mathematical modeling transforms real-world problems into mathematical representations. With modeling, you can analyze relationships, predict outcomes, and make decisions based on mathematical results.
The bridge toll problem uses a linear inequality model to answer practical questions about cost-effectiveness. Key steps in mathematical modeling include:

• Defining the problem and variables: crossing fee, pass cost, etc.
• Creating mathematical expressions or equations: cost functions for each scenario.
• Setting up equations or inequalities to compare options: formulating \(3n > 7.5 + 0.5n\).
• Solving to interpret outcomes: finding \(n > 3\) indicates use of a three-month pass is cheaper for more than three crossings.

Modeling helps bridge the gap between abstract math and everyday life decisions.